The convexity of \(\mathcal {X}\times \Lambda \)  ensures the Taylor expansion of \(\mathcal {L}_{\mu , \nu }\)  at any iterate.
The Lipschitz continuity of \(\nabla ^2\mathcal {L}\)  relaxes the thrice continuous differentiability of \(f, c\) , as assumed in [1]}.
Note that imposing conditions on \(\nabla ^2\mathcal {L}\)  is equivalent to imposing the same conditions on its components \(\nabla _{{x}}^2\mathcal {L}\)  and \(G({x})\) , although the Lipschitz constant may be different.
The Lipschitz continuity of \(\nabla f\)  is implied by the compactness of the set \(\mathcal {X}\) , while we express it out explicitly with constant \(\Upsilon _L\) . With the above setup and a simple calculation, we know from (REF ) that \(\nabla \mathcal {L}_{\mu , \nu }\)  is also Lipschitz continuous in \(\mathcal {X}\times \Lambda \) .
That is, for any \(({x}, {\lambda }), ({x}^{\prime }, {\lambda }^{\prime })\in \mathcal {X}\times \Lambda \) ,
\(\Vert \nabla \mathcal {L}_{\mu , \nu }({x}, {\lambda }) - \nabla \mathcal {L}_{\mu , \nu }({x}^{\prime }, {\lambda }^{\prime })\Vert  \\ \le {{1+(2\nu +\mu )\Upsilon _u}\Upsilon _u + (2\nu +\mu )\Upsilon _u\Upsilon _L} \Vert ({x}-{x}^{\prime }, {\lambda }-{\lambda }^{\prime })\Vert  \Upsilon _{\mu , \nu }\Vert ({x}-{x}^{\prime }, {\lambda }-{\lambda }^{\prime })\Vert .\) 
